Integrand size = 15, antiderivative size = 159 \[ \int \frac {x^{7/2}}{a+b x^2} \, dx=-\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{5/2}}{5 b}-\frac {a^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{9/4}}+\frac {a^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{9/4}}+\frac {a^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} b^{9/4}} \] Output:
-2*a*x^(1/2)/b^2+2/5*x^(5/2)/b-1/2*a^(5/4)*arctan(1-2^(1/2)*b^(1/4)*x^(1/2 )/a^(1/4))*2^(1/2)/b^(9/4)+1/2*a^(5/4)*arctan(1+2^(1/2)*b^(1/4)*x^(1/2)/a^ (1/4))*2^(1/2)/b^(9/4)+1/2*a^(5/4)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x^(1/2) /(a^(1/2)+b^(1/2)*x))*2^(1/2)/b^(9/4)
Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.81 \[ \int \frac {x^{7/2}}{a+b x^2} \, dx=\frac {4 \sqrt [4]{b} \sqrt {x} \left (-5 a+b x^2\right )-5 \sqrt {2} a^{5/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+5 \sqrt {2} a^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{10 b^{9/4}} \] Input:
Integrate[x^(7/2)/(a + b*x^2),x]
Output:
(4*b^(1/4)*Sqrt[x]*(-5*a + b*x^2) - 5*Sqrt[2]*a^(5/4)*ArcTan[(Sqrt[a] - Sq rt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 5*Sqrt[2]*a^(5/4)*ArcTanh[(S qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(10*b^(9/4))
Time = 0.42 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.56, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {262, 262, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^{7/2}}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \int \frac {x^{3/2}}{b x^2+a}dx}{b}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \int \frac {1}{b x^2+a}d\sqrt {x}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{b}\) |
Input:
Int[x^(7/2)/(a + b*x^2),x]
Output:
(2*x^(5/2))/(5*b) - (a*((2*Sqrt[x])/b - (2*a*((-(ArcTan[1 - (Sqrt[2]*b^(1/ 4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1 /4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[S qrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^( 1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt [2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/b))/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.32 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(-\frac {2 \left (-\frac {b \,x^{\frac {5}{2}}}{5}+a \sqrt {x}\right )}{b^{2}}+\frac {a \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2}}\) | \(125\) |
| default | \(-\frac {2 \left (-\frac {b \,x^{\frac {5}{2}}}{5}+a \sqrt {x}\right )}{b^{2}}+\frac {a \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2}}\) | \(125\) |
| risch | \(-\frac {2 \left (-b \,x^{2}+5 a \right ) \sqrt {x}}{5 b^{2}}+\frac {a \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2}}\) | \(126\) |
Input:
int(x^(7/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-2/b^2*(-1/5*b*x^(5/2)+a*x^(1/2))+1/4*a/b^2*(a/b)^(1/4)*2^(1/2)*(ln((x+(a/ b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b) ^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1 /4)*x^(1/2)-1))
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.05 \[ \int \frac {x^{7/2}}{a+b x^2} \, dx=\frac {5 \, b^{2} \left (-\frac {a^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (b^{2} \left (-\frac {a^{5}}{b^{9}}\right )^{\frac {1}{4}} + a \sqrt {x}\right ) + 5 i \, b^{2} \left (-\frac {a^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (i \, b^{2} \left (-\frac {a^{5}}{b^{9}}\right )^{\frac {1}{4}} + a \sqrt {x}\right ) - 5 i \, b^{2} \left (-\frac {a^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-i \, b^{2} \left (-\frac {a^{5}}{b^{9}}\right )^{\frac {1}{4}} + a \sqrt {x}\right ) - 5 \, b^{2} \left (-\frac {a^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-b^{2} \left (-\frac {a^{5}}{b^{9}}\right )^{\frac {1}{4}} + a \sqrt {x}\right ) + 4 \, {\left (b x^{2} - 5 \, a\right )} \sqrt {x}}{10 \, b^{2}} \] Input:
integrate(x^(7/2)/(b*x^2+a),x, algorithm="fricas")
Output:
1/10*(5*b^2*(-a^5/b^9)^(1/4)*log(b^2*(-a^5/b^9)^(1/4) + a*sqrt(x)) + 5*I*b ^2*(-a^5/b^9)^(1/4)*log(I*b^2*(-a^5/b^9)^(1/4) + a*sqrt(x)) - 5*I*b^2*(-a^ 5/b^9)^(1/4)*log(-I*b^2*(-a^5/b^9)^(1/4) + a*sqrt(x)) - 5*b^2*(-a^5/b^9)^( 1/4)*log(-b^2*(-a^5/b^9)^(1/4) + a*sqrt(x)) + 4*(b*x^2 - 5*a)*sqrt(x))/b^2
Time = 17.17 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.86 \[ \int \frac {x^{7/2}}{a+b x^2} \, dx=\begin {cases} \tilde {\infty } x^{\frac {5}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {9}{2}}}{9 a} & \text {for}\: b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 b} & \text {for}\: a = 0 \\- \frac {2 a \sqrt {x}}{b^{2}} - \frac {a \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {a \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {a \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} + \frac {2 x^{\frac {5}{2}}}{5 b} & \text {otherwise} \end {cases} \] Input:
integrate(x**(7/2)/(b*x**2+a),x)
Output:
Piecewise((zoo*x**(5/2), Eq(a, 0) & Eq(b, 0)), (2*x**(9/2)/(9*a), Eq(b, 0) ), (2*x**(5/2)/(5*b), Eq(a, 0)), (-2*a*sqrt(x)/b**2 - a*(-a/b)**(1/4)*log( sqrt(x) - (-a/b)**(1/4))/(2*b**2) + a*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)** (1/4))/(2*b**2) + a*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**2 + 2*x** (5/2)/(5*b), True))
Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.22 \[ \int \frac {x^{7/2}}{a+b x^2} \, dx=\frac {2 \, {\left (b x^{\frac {5}{2}} - 5 \, a \sqrt {x}\right )}}{5 \, b^{2}} + \frac {\frac {2 \, \sqrt {2} a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} a^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}}{4 \, b^{2}} \] Input:
integrate(x^(7/2)/(b*x^2+a),x, algorithm="maxima")
Output:
2/5*(b*x^(5/2) - 5*a*sqrt(x))/b^2 + 1/4*(2*sqrt(2)*a^(3/2)*arctan(1/2*sqrt (2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/s qrt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*a^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1 /4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt( b)) + sqrt(2)*a^(5/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sq rt(a))/b^(1/4) - sqrt(2)*a^(5/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sq rt(b)*x + sqrt(a))/b^(1/4))/b^2
Time = 0.12 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.23 \[ \int \frac {x^{7/2}}{a+b x^2} \, dx=\frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{3}} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{3}} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} a \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{3}} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} a \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{3}} + \frac {2 \, {\left (b^{4} x^{\frac {5}{2}} - 5 \, a b^{3} \sqrt {x}\right )}}{5 \, b^{5}} \] Input:
integrate(x^(7/2)/(b*x^2+a),x, algorithm="giac")
Output:
1/2*sqrt(2)*(a*b^3)^(1/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sq rt(x))/(a/b)^(1/4))/b^3 + 1/2*sqrt(2)*(a*b^3)^(1/4)*a*arctan(-1/2*sqrt(2)* (sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^3 + 1/4*sqrt(2)*(a*b^3)^( 1/4)*a*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^3 - 1/4*sqrt(2)* (a*b^3)^(1/4)*a*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^3 + 2/ 5*(b^4*x^(5/2) - 5*a*b^3*sqrt(x))/b^5
Time = 0.21 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.42 \[ \int \frac {x^{7/2}}{a+b x^2} \, dx=\frac {2\,x^{5/2}}{5\,b}-\frac {2\,a\,\sqrt {x}}{b^2}-\frac {{\left (-a\right )}^{5/4}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{b^{9/4}}+\frac {{\left (-a\right )}^{5/4}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}}\right )\,1{}\mathrm {i}}{b^{9/4}} \] Input:
int(x^(7/2)/(a + b*x^2),x)
Output:
(2*x^(5/2))/(5*b) - (2*a*x^(1/2))/b^2 - ((-a)^(5/4)*atan((b^(1/4)*x^(1/2)) /(-a)^(1/4)))/b^(9/4) + ((-a)^(5/4)*atan((b^(1/4)*x^(1/2)*1i)/(-a)^(1/4))* 1i)/b^(9/4)
Time = 0.20 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.99 \[ \int \frac {x^{7/2}}{a+b x^2} \, dx=\frac {-10 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+10 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-5 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )+5 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )-40 \sqrt {x}\, a b +8 \sqrt {x}\, b^{2} x^{2}}{20 b^{3}} \] Input:
int(x^(7/2)/(b*x^2+a),x)
Output:
( - 10*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a + 10*b**(3/4)*a**(1/4)*sqrt(2)* atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sq rt(2)))*a - 5*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*s qrt(2) + sqrt(a) + sqrt(b)*x)*a + 5*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)* b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a - 40*sqrt(x)*a*b + 8*sq rt(x)*b**2*x**2)/(20*b**3)